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Metamath Proof Explorer

Theorem syl2anb

Description: A double syllogism inference. (Contributed by NM, 29-Jul-1999)

Ref Expression
Hypotheses syl2anb.1 ⊢ ( 𝜑 ↔ 𝜓 )
syl2anb.2 ⊢ ( 𝜏 ↔ 𝜒 )
syl2anb.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion syl2anb ( ( 𝜑 ∧ 𝜏 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 syl2anb.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 syl2anb.2 ⊢ ( 𝜏 ↔ 𝜒 )
3 syl2anb.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4 1 3 sylanb ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
5 2 4 sylan2b ⊢ ( ( 𝜑 ∧ 𝜏 ) → 𝜃 )